A more careful and quantitative comparison between the model predictions and DNS data for \(Q,R\)  can be made by comparing the PDFs of \(Q\)  and \(R\)  separately. In Figure REF (a), one can see that in the DNS, the PDF of \(Q\)  is strongly positively skewed, which is associated with the fact that the vorticity is more intermittent than the strain-rate in turbulent flows [1]}. By contrast, the ML-RDGF model predicts PDFs for \(Q\)  that are much more symmetric, with the model significantly underpredicting regions of intense vorticity, and slightly underpredicting regions of intense straining. These discrepancies are, however, mainly in the tails of the PDF, with the model predictions in good agreement with the DNS for values of the PDF that are \(\ge O(10^{-2})\) .
